Wnumbers

Wnumbers are a hypothetical class of real numbers defined for a fixed positive integer W. A real number x in the unit interval is called a W-number (with parameters W and S) if its decimal expansion can be partitioned into consecutive blocks of length W such that the sum of the digits in every block equals a common constant S. Here S is an integer in the range 0 to 9W.

This definition yields a family of numbers for each pair (W,S). For example, with W=2 and S=5 one can construct x = 0.232323..., since the blocks "23" each sum to 5. More generally, any infinite sequence obtained by repeating blocks of length W whose digits sum to S is a W-number; nonrepeating constructions are also possible by varying blocks while preserving the per-block sum.

W-numbers form a countable set because blocks chosen from finite choices yield countable combinations. If the produced sequence repeats blocks, the decimal is eventually periodic; many W-numbers are thus rational. However, allowing nonperiodic variation within blocks yields non-repeating, non-rational examples. The concept stresses block-wise digit-sum constraints and is orthogonal to classical notions such as normality.

Variants include changing the base, letting the per-block sum vary by block with average S, or changing the constraint to sum modulo M; the idea is used in puzzles and in discussions of digit-constrained sequences in combinatorics on words. It is separate from, but related to, Champernowne-like constructions and normal numbers. See also Champernowne constant, normal numbers, digital sum, combinatorics on words.